# Modeling Snow Crystal Growth I: Rigorous Results for Packard's Digital Snowflakes

@article{Gravner2006ModelingSC, title={Modeling Snow Crystal Growth I: Rigorous Results for Packard's Digital Snowflakes}, author={Janko Gravner and David Griffeath}, journal={Experimental Mathematics}, year={2006}, volume={15}, pages={421 - 444} }

Digital snowflakes are solidifying cellular automata on the triangular lattice with the property that a site having exactly one occupied neighbor always becomes occupied at the next time step. We demonstrate that each such rule fills the lattice with an asymptotic density that is independent of the initial finite set. There are some cases in which this density can be computed exactly, and others in which it can only be approximated. We also characterize when the final occupied set comes within…

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## References

SHOWING 1-10 OF 57 REFERENCES

### Cellular Automaton Growth on Z2: Theorems, Examples, and Problems

- Physics
- 1998

We survey the phenomenology of crystal growth and asymptotic shape for two-dimensional, two-state cellular automata. In the most tractable case ofThreshold Growth, a detailed rigorous theory is…

### The physics of snow crystals

- Materials Science
- 2005

We examine the physical mechanisms governing the formation of snow crystals, treating this problem as a case study of the dynamics of crystal growth from the vapour phase. Particular attention is…

### Physics of crystal growth

- Materials Science
- 1998

Preface List of symbols 1. Morphology of a crystal surface 2. Surface free energy, step free energy, and chemical potential 3. The equilibrium crystal shape 4. Growth and dissolution crystal shapes:…

### Universality in Elementary Cellular Automata

- Computer ScienceComplex Syst.
- 2004

The purpose of this paper is to prove that one of the simplest one dimensional cellular automata is computationally universal, implying that many questions concerning its behavior, such as whether a…

### Snow Crystals

- PhysicsNature
- 1901

AFTER the recent heavy snow in this district, the slight fall yesterday afternoon did not, at first, attract much attention, appearing like sleet to the casual observer. It proved, however, to be of…

### A new kind of science

- Computer Science
- 2002

A New Kind of Science, written and published by Stephen Wolfram, is the outcome of the studies he conducted systematically upon cellular automata, a class of computer model which may be visualized as a set of memory locations, each containing one bit.

### Fractals, scaling, and growth far from equilibrium

- Physics
- 1998

This 1998 book describes the progress that had been made towards the development of a comprehensive understanding of the formation of complex, disorderly patterns under conditions far from…

### Computation theory of cellular automata

- Computer Science
- 1998

The sets of configurations generated after a finite number of time steps of cellular automaton evolution are shown to form regular languages and it is suggested that such undecidability is common in these and other dynamical systems.

### Hausdorff dimension in graph directed constructions

- Mathematics
- 1988

We introduce the notion of geometric constructions in Rm governed by a directed graph G and by similarity ratios which are labelled with the edges of this graph. For each such construction, we…